Flexural vibrations of rotors immersed in dense fluids Part II: Experiments

Journal of Fluids and Structures (1992) 6, 23-38

FLEXURAL

VIBRATIONS OF ROTORS IMMERSED IN DENSE FLUIDS PART II: EXPERIMENTS

J. ANTUNES,* F. AXISA,** F. HAREUX** *Nuclear Energy and Engineering Department, Laboratorio National de Engenharia e Techno Logia Industrial 2686 Sacavkm Codex, Portugal ** Dkpartement des Etudes Mkaniques et Thermiques, Centre d’Etudes Nuclkaires de Saclay Commissariat d 1‘Energie Atomique, 91191 Gif-sur-Yvette Cedex, France

(Received 12 January 1990 and in revised form 28 January 1991) This paper describes experimental work carried out on a homogeneous rotating shaft partially immersed in an annular space of water. Experimental results concerning the natural frequencies and reduced damping of the first pair of transverse modes are found to agree fairly well with the analytical model presented in Part I, at least in a rotating velocity range below the buckling instability of the retrograde mode. 1. INTRODUCTION

IN PARTI OF THIS PAPER,a general formulation for the vibratory behaviour of a rotating shaft under dense fluid confinement has been presented (Axisa & Antunes 1991), including fluid effects such as added mass, viscosity and fluid friction. In Part II (this paper), an extensive experimental programme is reported, enabling the comparison between theory and test results, in order to validate the analytical approach. Theoretical papers on the subject are scarce, and the same is true concerning experimental results. Following the paper by Taylor (1936), several other authors studied the characteristics of fluid flow between coaxial rotating cylinders. However, interest was focussed on fluid dynamics and not on structural vibrations. In the experimental paper by Fritz (1970), the vibratory response of a flexible rotor under fluid confinement was tested. His test rig was somewhat similar to the one used in the present work. However, the scope of the tests described in Fritz (1970) is rather restricted, in the sense that no attempt was made to measure the modal frequencies and damping ratios of both forward and backward whirl modes (also known as direct and retrograde modes, respectively). Most of the results presented concern the response of the rotating shaft to synchronous unbalance forces. Other pertinent work is reported in papers by Ramsden et al. (1974, 1975), in the context of fast reactor sodium pumps design. The last reference is concerned with experimental work performed on primary sodium pumps, including measurements on site. Again, attention is focussed mainly on the synchronous response due to unbalance. Therefore, it seems that the present investigation could narrow a serious gap in the open literature, concerning the vibratory problem under consideration. Specifically, the test results reported here will address the following issues: l

l

identification of the rotor forward and backward whirl modes, under fluid confinement; establishment of the relevant trends for modal frequencies and damping ratios as functions of the shaft spinning velocity and of the fluid level;

0889-9746/92/010023 + 16 $03.00

0

1992 Academic Press Limited

J. ANTUNES ET AL

24

identification of the dominant physical phenomena, experimental results and theoretical predictions. l

and comparison

between

As will be seen, the experimental results obtained are in reasonable agreement with the theory developed in Part I. In particular, analytical predictions of modal frequencies are in excellent agreement with experiment. 2. EXPERIMENTAL 2.1.

TEST

APPARATUS

MODEL

The test rig is shown in Figure 1. It consists of a hollow cylindrical steel shaft with 75 mm external radius, 4 mm thick, l-5 m long, and with an overall mobile mass of about 50 kg. The shaft was assembled with its axis vertical, maintained at the upper end by a support plate, through conical bearings. Translational and angular support

Figure

1. Test rig

VIBRATIONS OF ROTORSIN FLUIDS:Fr II

25

stiffness is provided by eight steel struts, equally spaced around the shaft, linking the rotor support plate to the main structure bolted to the laboratory floor. The basic modal characteristics of the rotor could be conveniently adjusted using several sets of struts with various geometries. However, all tests reported here pertain to an unique geometrical configuration. The shaft rotates inside a coaxial annular gap, filled with water up to a height of 1 m, with a clearance of 11 mm between the centered shaft and the rigid outer shell. Steady rotational shaft motion is provided by an electric motor, controlled by velocity feedback, ranging from zero up to 700rpm. Torque transmission to the rotor is achieved through a flexible double cardan joint, providing negligible additional transverse stiffness. The mobile fixture is excited laterally by a 200 N electromechanical shaker, at the support plate level, delivering a random force of band limited while noise in the range O-125 Hz. Force transmission between the shaker and the rotor is achieved through a small rod, providing negligible coupling between axial and transverse motions. 2.2. IN~TRuMBNTA~I~N The location of various transducers used in the tests is shown in Figure 2. A pair of orthogonal accelerometers, as well as a pair of noncontacting inductive displacement sensors are located in the upper support plate. These were used for the main body of rotational tests. Four additional accelerometers, located at the lower end of the shaft and on the external wall of the outer cylindrical shell, were used in preliminary tests for identification of the shaft and structural modes. Monitoring of the excitation random force was provided by a piezoelectric transducer, located in the force transmission path. Analog signals delivered by the various transducers were filtered, conditioned, amplified and digitized, before recording and analysis. Real-time signal monitoring was achieved using a four-channel scope and an HP 3582 two-channel spectral-analyzer. Subsequent transfer of digitized signals into an HP 9816 microcomputer enabled the identification of modal parameters. Analysis of long-term transient responses was performed using an eight-channel Gould strip-chart recorder/digitizer. 2.3. EXPERIMENTAL PROCEDURE The preliminary tests were performed in air and quiescent water, for several values of the annular column height in the range O-1.0 m. Furthermore, for the rotational tests, spinning velocities up to 600 rpm were used. Identification of modal parameters were based on the rotor response spectrum to white noise force excitation. Modal frequencies and damping ratios of the forward and backward whirl modes were estimated using an interactive computer program, based on a multi degree-of-freedom fitting algorithm in the frequency domain, Pettigrew et al. (1985). It is worth mentioning that all pertinent modes of the rotor are well within the O-125 Hz range of the white noise excitation, as will be shown later in this paper. Response spectra were obtained after averaging the FFTs of 128 time-history acquisitions. Use of the analyzer “zoom” feature enabled spectral resolution up to Af If; - W3, where f; stands for each modal frequency. Hanning windowing was adopted for time-weighting the system response signals. The amplitude of synchronous unbalance response was monitored, as well as the corresponding critical velocities. Vibration amplitudes were computed through integration of the response spectrum in the vicinity of the modal frequencies. Finally some interesting observations were recorded, concerning the co-rotational fluid flow, specially near the free surface level.

26

J.ANTUNES

ETAL.

Electrical Douhlr

motor

cardan joint -----, Force transducer

7 ’ ACC-H

Shaft --

i

1

kJL

_ =+q--

Figure

2. Model

layout

Water tanh Air suppi>

and instrumentation

3. TEST RESULTS 3.1. PRELIMINARY TESTS WITH SHAFT AT REST The natural frequencies of the shaft first modes were measured, in air, at 10.7 Hz and 23-4 Hz. Further tests confirmed the essentially linear behaviour of the fixture, and showed the modal response of the supporting structure to be sufficiently decoupled from the shaft, in the frequency range of interest. In all cases, flexural deformation of the shaft was found to be negligible. The rotor operates essentially as a rigid body, and modal frequencies stem from the geometry of the flexible supporting struts, as described in Section 2.1. The amplitude of the shaft upper end is, for the first mode, about half the corresponding value at the lower end. The second mode of the model corresponds to a rocking motion of the shaft.

VIBRATIONS OF ROTORS IN FLUIDS:PT II

27

TABLET

The values of first-mode frequency, f,, and damping, [,, for various values of H.

0.8

1.5 2.0 2.2

10.7 6.8 5.4 4.9

0 0.3 O-6 0.9

The frequency of the first mode was measured in both orthogonal directions, showing a 2% variation, which points to a small asymmetry in the support stiffness. Structural damping of 0.8% was measured in air. In still water, added mass and viscosity effects produced significant changes in modal parameters, as shown in Table 1, for various values of the annulus height. A polynomial regression fit of the complete measurement set shows that frequency decreases as a parabolic function of H, whereas damping increases in a linear fashion.

3.2. VIBRATORY

RESPONSE OFTHE

ROTATING

SHAFT

Figure 3 shows significant changes in the rotor first mode response, when the spinning velocity increases from zero up to 400 rpm, for a value H = O-2 m of the annulus height. The split of the non-rotating shaft first natural frequency into two response peaks is clearly seen. The separation between those peaks increases with the rotor velocity, as should be expected from theory (Axisa & Antunes 1991). The sharp peak, which can be seen in the third response spectrum of Figure 3, corresponds to the forced shaft response to synchronous unbalance forces. Figure 3(d) shows this peak under different operating conditions (velocity 300 rpm, H = 600 mm). The first harmonic of unbalance response, which can also be seen, is indicative of some nonlinearity of the coupled fluid-structure system. Analysis of the phase relationship between the responses of two orthogonal transducers confirmed that the higher frequency peak corresponds to forward whirl motion, whereas the lower peak corresponds to backward motion, in accordance with theory. Damping measurements indicate that both forward and backward whirl modes are more heavily damped than the original non-rotating shaft mode. It must be emphasized that the behaviour previously described was not observed during the rotational tests in air, and the frequency split did not occur for the complete range of test velocities. This confirms that structural gyroscopic effects are negligible and that the rotor response is dominated by fluid forces, as discussed in Part I. Quantitative results concerning the change of modal frequencies with the spinning velocity and annulus height are shown in Figure 4. The following effects can be seen: (i) no significant change in modal response with spinning velocity is observed in air tests, as stated before; (ii) a decrease of all modal frequencies, due to fluid added mass, which is greater for higher values of the liquid level; (iii) an essentially linear increase of the forward mode frequency with rotational speed, as well as a linear decrease of the backward mode frequency, both slopes being almost

J. ANTUNES

(a)

!‘v = 0 rpm.

H = 3HImm

Frequent!

(cl

N = 4gO rpm.

E7’ AL

( 131

Y = 3Hl rpm. I{ = 21HI mm

( HI)

H = 2(M) mm

6.38

1 I.56 I.requenq Figure 3. Rotor

( HL)

response spectra as a function

f:rcqwncy of spinning velocity

(Hr)

and water level

symmetric and nearly insensitive to the liquid level, at least when H > 0.3 m. It can also be seen, from Figure 4, that the velocity range of the present tests was insufficient to reach the value where the backward mode frequency becomes zero. The change of modal damping ratios, with the shaft velocity and the liquid level, is shown in Figure 5. Large damping measurements are difficult to perform, and the situation can be further complicated when modal peaks are in the vicinity of the sharp unbalance response peak. Those factors certainly contribute towards the somewhat less than adequate scattering of damping measurements. Nevertheless, some general statements can be drawn from Figure 5, as follows: (i) during air tests, damping variations remained comparatively negligible; (ii) damping increases with the rotor velocity and the liquid column height, for both forward and retrograde whirl modes; (iii) damping of the backward mode is always greater than the corresponding value of the forward mode (values as large as 16% were measured in the present tests).

VIBRATIONS OF ROTORS IN FLUIDS:I-Y II

29

Forward whirl mode

Rotor Lelocit!,. Backward

:V, ( rpm)

whirl mode

0 650.0

(141 Rotclr vrlocttl;. ,t’, (rpm)

Figure 4. Forward and backward modal frequencies as a function of rotor velocity and water level. (a) Forward whirl mode (critical velocities refer to the intersections of the direct mode curves with the unit slope n , H=3OOmm; +, H=6OOmm; ., H=9OOmm; line; (b) backward whirl mode. 0, H=O; 4, H= l,OOOmm.

3.3. CRITICAL VELOCITIES OF UNBALANCE RESPONSE Critical showing

spinning velocities

are conveniently

the change of the forwurd

modal

extracted frequency

from

the curves of Figure

with rotor

velocity

4,

and liquid

level. Synchronous response is given by the crossing of the unit-slope straight line with each one of the frequency evolution curves. Some figures thus obtained are shown in Table 2. N, refers to the critical value of NR. The change of the critical velocity with the liquid level is almost parabolic. This is to be expected, when the results previously described are accounted for. The values of to the various critical velocities, were displacement amplitudes, 6,, corresponding measured at the support plate level. Response amplitudes away from the critical velocities were one order of magnitude lower than at N,.

30

J.ANTUNES

Rotor

velocity,

El‘ AL

NH (rpm)

h5ll.(I

Rotorvelocity. Nn (t-pm) Figure 5. Forward and backward modal damping ratios as a function of rotor velocity and water level. (a) Forward whirl mode; (b) backward whirl mode. 0, H = 0; n , H = 300 mm; +. H = 600mm; b, H=!%Omm; 4, H= l.OOOmm.

4. COMPARISON

WITH THEORETICAL

PREDICTIONS

4.1. VIBRATION OF THE NON-ROTATING SHAFT The theoretical principles developed in Part I will now be applied to the geometry and physical parameters of the test model. The shaft is modelled in terms of a distributed mass per unit length, m (total length, L) and two point masses, M, and M2

VIBRATIONS

OF ROTORS

IN FLUIDS:

PT II

31

TABLE 2 Critical angular velocities, N,, and the corresponding vibration amplitudes, a,, for various values of H. 4. (ml -400 -240 -190

N (wd 625 540 490 420

H (ml 0 0.3 0.6 0.9

at the shaft ends (X = L and x = 0). The shaft support is conveniently modelled through equivalent translational stiffnesses, KY = K, = K, and angular stiffnesses, k, = k, = k. These are related to the first mode frequency and modal shape in air:

(1) Then, from the physical parameters, M, = 25.3 kg, M2 = 1.9 kg, m = 14.3 kg m-‘, L = 1.47 m, and accounting for the experimental measurements of o, and V,(X), the values K = 2.7 x lO’Nm_’ and k = 2.1 x lo’Nm/rad are obtained. The generalized masses are, for the first two shaft modes in air, A?, = 20.5 kg and &ZZ= 14.5 kg. In still water, the shaft modes are modified by the fluid,

(2) with the generalized fluid forces H

2n

$j=-

p(R,, 8, x)R, cos hp&) de dr, (3) II where the pressure field is g;vei by Laplace equation V’p = 0. The boundary conditions applying to the fluid are ill defined at the lower end of the shaft, where the annulus communicates with a water tank. Three-dimensional calculations, for different conditions, showed that the system vibrates essentially as if the lower end of the fluid annulus was closed by a rigid wall. However, the actual behaviour depends on both the water level and the rotor velocity. Postulating a closed annulus bottom, and accounting for the effect of free surface, the generalized inertial forces

(4) are given by Axisa & Antunes (1991), Mij = mf

1 _ cosh(xlR)

cosh(H/R)

1

&h+)

d&

R3 mf =-pn. ho

(5)

In the previous formulation, gravity and compressibility effects were safely neglected, as discussed in Part I. The coupled fluid-structure eigenvalues, obtained from

32

J. ANTUNES ET

AL

TABLE 3 Comparison between experimental and computed eigenfrequencies for various values of H.

H

equations

(1,5),

(m)

Test fi 0-W

Theory fi (Hz)

k, (kg)

0 0.3 0.6 0.9

10.7 6.8 54 4.9

10.7 6.8 5.3 4.6

20.5 40.6 67.2 87.1

are the roots of

[M,f$ - (M, + M,$P][M*W:

- (& + M&P]

- M$P

= 0.

(6)

Tabie 3 shows the measured and computed frequencies of the first mode, as we11 as the computed generalized mass. If one accounts for the above-mentioned uncertitude in the fluid boundary conditions, these results seem adequate. For the non-rotating shaft, viscous fluid damping can be conveniently estimated using the asymptotic formulation [see part I, equation (26)], (7) or from the more involved formulation used in the computer code ROTOR [Part I, equation (22)]. The experimental viscous damping. cy, is estimated from the total measured damping, CT, in the following way:

L(W = L-T(H)- 5, (()I.

(8)

Under test conditions. fluid viscosity is v = Il)Yh m’/s. In Table 4, the various experimental and computed values are compared. From these values it is clear that ROTOR estimations, always within 25% of the

measured values, are better than those obtainable from equation (7). Also, it is important to note that viscosity induces additional damping which is small when compared with that observed during the rotational tests. TABLE 4 Total damping,

H

CT, and viscous damping, various values of H.

Measurements

(m)

5,(R)

0 o-3 0.6

0.8 1.5 2.0

0.9

2.2

Equation

(7)

<,. for

ROTOR

L (%i

L (S,

i” (S)

0 0.7

0 0.62

0 0.73

1.2 1.4

0.87 0.98

1.05 1.11

33

VIBRATIONS OF ROTORS IN FLUIDS:PT II

4.2.

RESPONSE OF THE ROTATING

SHAFT

As shown in Section 3.2, and in agreement with the discussion of Part I (Section 2.2), structural gyroscopic coupling effects on the test model are negligible. Therefore, the rotor dynamics are described in terms of the following system (Axisa & Antunes 1991): M,+M, 0

0 Ms +Mo

I( 1 ii,

+

iiz

A,+A,+A,

M,Q

-M&J

A, +A,

MS&-

+

+A,

Ma+2

+ -Kf

where A, = 2MsooJ-0,

A,, = M,[2vw,/h;]‘“,

and &, is the structural modal damping in air, w. is the modal circular frequency in air, o, is the real part of the coupled system response frequency, MS is the structural modal mass, and M, is the modal added mass of the fluid. The eigenvalues of the coupled system (9) are then given, for each value of the spinning velocity, Q, by the complex roots of

m2-

M,toz - h4aF2 - K,i

M,Q + Ai lG

w-

M

=

0

(10)

for forward modes, and by m2 _ -M,B

h4,ui - Ma:

+ Ai El

o-

&l

+ K,i =

0

(11)

for the backward modes (in the previous equations %#= MS + M, and A = A, + A,, + response circular frequencies are calculated as I$,~ = ,%( w&, and reduced damping is given by &-p*”= [~%,,/(l + K%,~)]~", where ~~~~ =

A, ). Therefore,

$~(OLI,R)~WOD,R)-

As discussed in Part I, the skin-friction coefficient, f, is an important parameter the previous formulation. For turbulent flow (Re > 4000), f can be calculated as f=$

1.14 - 2 log,,

in

(12)

where Re = &RD,I2v and DH = 2h, (Blevins 1984). Table 5 shows the values thus obtained, for several values of the spinning velocity and the wall roughness, E. It is clear that the hypothesis of turbulent flow is consistent with the experimental conditions (in fact, Re > 4000 for velocities as low as 46 rpm). Furthermore, Table 5 shows that f is nearly independent of N and moderately dependent on E, in the velocity range of interest. From Blevins (1984), the wall roughness of steel pipes ranges from 0.02 up to 1 mm, according to surface conditions, leading to skin friction coefficients in the range

34

J. ANTUNES

ET AL

TABLE5 Skin friction coefficient as a function of Reynolds number and wall roughness, e. E (mm) (qfn) 100

300 500

Re

__..0.1

0.25

0.012 2.6 x 10” 0.008 0.011 4.3 x lo4 OGXj 0.010 86 x IO3 0.010

0.5

1

O-014 0.018 0.013 0.017 0.013 0.017

shown before. For the tests reported here, the value f = 0.013 produced the best correlation between experimental and analytical results. With such an order of magnitude, for the velocity range explored, modal frequencies are in no way affected by the skin-friction coefficient. In Figure 6, the forward and backward whirl test frequencies are compared with analytical results, which are identical from either formulation (9-11) or ROTOR finite element calculations. Agreement is, in general, excellent and theoretical predictions lay within 6% of the experimental results. The small difference between experimental and theoretical results is interpreted as a consequence of the nonuniform and very turbulent rotating flow near the surface level. During rotational tests, the fluid surface is not perpendicular to the spinning axis for all velocities greater than 200 rpm. When NR > 3OOrpm, a surface emulsion and waves can be distinctly seen, as shown in Figure 7. All those fluid effects are not accounted for, in the theoretical model. In Figure 8. the forward and backward modal damping ratios are also shown. In spite of the considerable scattering of the test values, there seems to be a reasonable qualitative agreement between experimental and analytical results. This agreement is better for high values of the water level, when free surface effects are comparatively less important. However, damping measurements in the velocity range about 100 rpm are always greater than theoretical predictions. It is interesting to note that very significant values of modal damping were obtained, in particular for the backward whirl mode, with measured values up to 16%. As stated in Part I, these figures cannot be explained in terms of viscosity effects. In fact, viscosity-induced effects remain rather low, a result in accordance with Fritz’s observations. Furthermore, if these effects were dominant, they would give rise to damping evolution trends completely opposite to those obtained in the experiments. The theoretical values of critical velocities are slightly higher (~5%) than the corresponding measurements. This is a direct consequence of the higher than measured frequency values for the direct mode, as predicted from theory.

5. CONCLUSION

An extensive test program has been presented, which provided results in reasonable agreement with theoretical predictions. In particular, excellent agreement has been found for the frequency of the forward and backward whirl modes. Experimental values of the corresponding damping ratios are rather scattered. However, qualitative

VIBRATIONS

OF ROTORS

35

IN FLUIDS: FI- II

2-

(a)

0:. 0

100

200

300

Rotor velocity.

400

So0

101

0 t 0

60

N, (rpm)

I

I 100

200

300

Rotor velocity.

400

f;(X)

MM)

N, (rpm)

IO

H = 000mm

Rotor velocity.

N, (rpm)

Figure 6. Comparison between theoretical and experimental 5;, = O.fj%, Y = 10e6 m’/s, f = 1.3 x 10e2. 0, Direct (forward mode.

modal frequencies. For the analytical whirl) mode; 0, retrograde (backward

model, whirl)

36

J. ANTUNES ET

Figure

7. Visualization

of the free-surface

AL

at N, = 300 rpm and

H = 900mm

agreement between theory and experiment can be found, provided that skin-friction effects are accounted for. It is interesting to note that, at higher spinning velocities than those of the present experiments, large amplitude subsynchronous shaft whirling motion was observed. That type of rotor response, which emerges at shaft velocities near the value Q, (where frequency of the backward mode becomes zero), will be discussed in a future paper.

VIBRATIONS

OF ROTORS

IN FLUIDS:

10

.

I

,\’ / ,

h-

2 2 ‘C

E

/

/

,

I

/

/

/ /

_ 0

4-

I

/

37

II

0

H = XHImm

x-

.s

I7

:

0

.

/’

/

0,’

.

Rotor velocity. NH (rpm) 10

/ H = 600mm

//

/

8

/ 0

(h) 0 ,,,,,,

0

,‘I

..,,,,,,I,

100

200

300

Rotor velocity.

.:

s

400

so0

f

500

6

N, (rpm)

//

12

2

F ._ 2

,,I-

.,.,.,,,‘..,,..‘,,,,,

0

8

//

//

/

//

4

0 0

loo

200

300

Rotor velocity. Figure ~=O+%,

400 Nx (rpm)

8. Comparison between theoretical and experimental damping ratios. For the analytical Y= 10 m*/s, f= 1.3 x 10m2. W, Direct (forward whirl) mode; El, retrograde (backward mode.

model, whirl)

ACKNOWLEDGEMENTS The experiments reported in this paper were performed with the very valuable assistance of Mr A. Jarrige, of the CEA/DEMT (Saclay). The authors are grateful to Professor R. A. Scott, who reviewed the paper and improved the English.

38

J.ANTUNES

ETAL

REFERENCES F. & ANTUNES, J. 1991 Flexural vibrations of rotors immersed in dense fluids: Part 1. Theory. Journal of Flu& and Structures, 6, 3-21. BLEVINS, R. D. 1984 Applied Fluid Dynamics Handbook, New York: Van Nostrand Reinhoid. FRITZ, R. 1970 The effects of an annular fluid on the vibrations of a long rotor: Part 2-test. ASME Journal of Basic Engineering 92, 930-937. PE-~TIGREW, M., TROMP, J. & MASTORAKOS, J. 1985 Vibration of tube bundles subjected to two-phase cross-flow. ASME Journal of Pressure Vessel Technology 107, 335-343. RAMSDEN, J., RITCHIE, G. & GUPTA, S. 1974 The vibrational response characteristics of a design for the sodium pumps of the commercial fast reactor. Paper C107/74. in Proceedings I. Mech. E. Fluid machinery and nuclear energy groups convention: pumps for nuclear power plant, pp. 187-196, Bath, April 1974. London: I. Mech. E. RAMSDEN, J., JONES, H. & COWKING, E. 1975 Vibration of the P.F.R. primary sodium pumps. Paper C103/75, in Proceedings I. Mech. E. Vibrations and noise in pump, fan and compressor installations, pp. 21-33, Southampton, September 1975. London: I. Mech. E. TAYLOR, G. I. 1936 Fluid friction between rotating cylindersProceedings of the Royal Society of London, Series A, l57, 546-564. AXISA,